We study the nonlinear stochastic diffusion equation of Fick’s type in which the porosity ɛ fluctuates strongly along the column axis. Green’s function for the equation of nonlinear diffusion is constructed for the case of Gaussian distributed fluctuating porosity. Investigating the properties of the theory, we have justified the previous results on the inclusion of a „time-diffusive„ term into the equation of nonlinear diffusion in case of fluctuating porosity for arbitrary dimension of space and for arbitrary stochastic process originating the porosity fluctuations. In particular, we formulate the phenomenological theory of transport through porous media, consider the problem of critical scaling, and justify the Kolmogorov and Richardson empirical laws from the microscopic equations. We calculate long-time, large-scale asymptotes for the Correlation function G and extinction coefficient X. Furthermore, we discuss critical dimensions of composite operators (time local averages of fields, their powers, and derivatives) and the solutions for arbitrary porosity. The relevant critical indices meet the empirical laws of Kolmogorov and Richardson for any porosity value. However, the quantities that are not universal, i.e., the amplitudes of empirical laws, are dependent on certain initial values of the parameters as well as porosity fluctuations. The fluctuating porosity forms the damping time spectrum for the dynamic correlation functions. Finally, the solution of a nonlinear equation of diffusion for the concentration field u(x, t) is a travelingwave with an amplitude damping in space and time. This fact distinguishes the theory considered from the case of ordinary nonlinear diffusion as well as the case of turbulent mixing of passive advection.
|Title of host publication||Focus on Porous Media Research|
|Publisher||Nova Science Publishers, Inc.|
|Number of pages||22|
|State||Published - Jan 1 2013|